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## Comment on

Triangles - Part II## At 7:21, if we draw an

## Good question.

Good question.

The answer is no. In order for the diagonal to bisect the top angle, the non-equal side must be the base.

Try it with a 45-45-90 triangle, and you'll see what I mean.

## Hi Brent,

A little confused with this. Suppose if the isosceles triangle is ABC with sides AB=AC, then will the altitude drawn from angle A bisect angle BAC in two equal parts? I thought it is only a bisector segment dividing side BC in two equal parts.

Thanks

## Yes, the altitude drawn from

Yes, the altitude drawn from point A WILL bisect angle BAC in two equal parts.

Think of it this way (using the diagram you describe with point D representing the point where the altitude intersects side BC)

We get two triangles: ∆ADB and ∆ADC

First, recognize that ∠ABD = ∠ACD, since those are the equal angles in ∆ABC

Second, recognize that ∠ADB = ∠ADC = 90°, since AD is the ALTITUDE

Third, since triangles ∆ADB and ∆ADC already have two angles in common, the third angles must also be equal. That is ∠BAD = ∠CAD

Does that help?

Cheers,

Brent

## Amazing. Without an inkling

Is a bisector drawn always perpendicular to the line it bisects or this is the property applicable only in case of Isosceles and Equilateral triangle?

Thanks.

## The bisector is perpendicular

The bisector is perpendicular ONLY IF the triangle is either isosceles and equilateral.

## Sir is there any easy

https://gmatclub.com/forum/in-the-figure-above-point-o-is-the-center-of-the-circle-and-107874.html

## Have you seen Karishma's

Have you seen Karishma's solution: https://gmatclub.com/forum/in-the-figure-above-point-o-is-the-center-of-...

If you have any questions about her solution, let me know and we can go from there.

## sir, i am having doubt in

https://gmatclub.com/forum/in-triangle-abc-point-x-is-the-midpoint-of-side-ac-and-82671.html

## I have provided a detailed

I have provided a detailed response here: https://gmatclub.com/forum/in-triangle-abc-point-x-is-the-midpoint-of-si...

## One of the statements from

"(1) x^2 + y^2≠ z^2"

Does this statement translate into the triangle being an equilateral triangle?

## Question link: https:/

Question link: https://gmatclub.com/forum/is-the-area-of-the-triangular-region-above-le...

IF it were the case that x² + y² = z², then we'd know that the triangle is an RIGHT TRIANGLE with z as the hypotenuse.

Since we're told that x² + y² ≠ z², then we know that z is not the hypotenuse of a right triangle.

ASIDE #1: This does not necessarily mean that the given triangle is not a right triangle. It could be the case that side x is the hypotenuse of the triangle, in which case, we'd get y² + z² = x²

ASIDE #2: It COULD also be the case that the given triangle is an equilateral triangle, but it could also not be an equilateral triangle.

Does that help?

Cheers,

Brent

## Hi Brent

Helps perfectly thank you, I saw in a later video this topic is discussed in more depth.

Regards

## Hi Brent,

Could you please present your answer on this?

https://gmatclub.com/forum/in-triangle-abc-point-x-is-the-midpoint-of-side-ac-and-82671.html

The question introduces properties of Midsegements, something i haven't come across in our course module.

Thanks.

## You bet!

You bet!

Here's my step-by-step solution: https://gmatclub.com/forum/in-triangle-abc-point-x-is-the-midpoint-of-si...

Cheers,

Brent

## Question Link: https:/

Would you mind explaining the solution. Specifically I would like to know the approach on how the deduce or crack the given info. I seem to understand but it when encountered again, I can't get it right

## Hi emailme,

Hi emailme,

Here's my step-by-step solution: https://gmatclub.com/forum/in-the-figure-above-point-o-is-the-center-of-...

Cheers,

Brent

## Hi,

In the following question, how did we figure out the height of the triangle?:

https://gmatclub.com/forum/given-the-three-points-below-what-is-the-area-of-the-triangle-formed-238212.html

Thanks!

## Hi aanchal890,

Hi aanchal890,

Link to my solution: https://gmatclub.com/forum/given-the-three-points-below-what-is-the-area...

First notice that the red base is on the line y = -3

We know this because the y-coordinates on the base are -3.

Next, notice that the y-coordinate of the top vertex is 3, since the coordinates are (0, 3)

So, the height = 3 - (-3) = 6

Does that help?

Cheers,

Brent

## Yes! Thank you for the

## Sir, could you please solve

https://gmatclub.com/forum/in-the-figure-point-d-divides-side-bc-of-triangle-abc-into-segments-126934.html

Thank you.

## Here's my step-by-step

Here's my step-by-step solution: https://gmatclub.com/forum/in-the-figure-point-d-divides-side-bc-of-tria...

Cheers,

Brent

## Thank you, sir, for the lucid

## Hi Brent, need your support

https://gmatclub.com/forum/what-is-the-area-of-a-triangle-created-by-the-intersections-158727.html

## Question link: https:/

Question link: https://gmatclub.com/forum/what-is-the-area-of-a-triangle-created-by-the...

My full solution: https://gmatclub.com/forum/in-the-past-most-children-who-went-sledding-i...

Cheers,

Brent

## Hi Brent, the question below

https://gmatclub.com/forum/in-the-figure-point-d-divides-side-bc-of-triangle-abc-into-segments-126934.html

## My mistake. The question

My mistake. The question largely tests one's knowledge of Special Right Triangles. So, I have moved the link to the video lesson that covers this concept: https://www.gmatprepnow.com/module/gmat-geometry/video/870

Cheers,

Brent

## Is the reverse true for both

I mean if the line bisects the base then does it also imply the line is perpendicular? or is it only the other way round.

## That's correct.

That's correct.

If we have an isosceles or equilateral triangle, and a line drawn from a vertex bisects the opposite side, then line is also perpendicular to the opposite side.

ASIDE: Just to be clear, for isosceles triangles, the side that is bisected must be the side that is NOT one of the two equal sides.

Cheers,

Brent

## Hi Brent,

In the question below, my answer is D. Here's how.

https://gmatclub.com/forum/in-the-triangle-shown-y-z-if-j-x-k-what-is-the-least-value-of-263317.html

The question is asking to calculate least possible value of k, which is greater than x.

Stmt 1: y=5 and since z>y, z could 6 and therefore x could be in the range of 1 to 11, making the least possible value of k as 1

Adopted a similar approach for stmt 2.

Where am I wrong here?

## Question link: https:/

Question link: https://gmatclub.com/forum/in-the-triangle-shown-y-z-if-j-x-k-what-is-th...

That's a good idea, but since we don't know the value of z, we can't make any conclusions about x.

For example, if z = 1000, then x cannot equal 1.

The same applies to statement 2.

Does that help?

Cheers,

Brent

## But isn't that in order to

## I should note that it's

I should note that it's EXTREMELY rare to have a Data Sufficiency question featuring a max/min component. In fact, I'm not sure I've actually seen an official GMAT question with a similar construction.

True be told, I was initially going to argue that the question was vulnerable to ambiguity.

Here's the dilemma: When we're given the lengths of TWO sides of a triangle, we can easily determine the range of possible lengths for the 3rd (unknown) side. However, what conclusions can we draw when we know the length of just ONE side (as is the case with this DS question)?

Statement 1 tells us that y = 5, so it's possible that z = 5.00000000001, in which case, the length of the 3rd side must be greater than 0.00000000001.

So, k can actually be less than 1 (notice that we aren't told that any of the values are integers).

Ultimately, I think the question inherently flawed (and un-GMAT-like)

Cheers,

Brent

## Hi Brent,

Need your help:

A triangle has side lengths of a, b, and c centimeters. Does each angle in the triangle measure less than 90 degrees?

1. The 3 semicircles whose diameters are the sides of the triangle have areas that are equal to 3 cm2, 4 cm2, and 6 cm2, respectively.

2. c < a + b < c + 2

## Another killer question!

Another killer question!

My solution: https://gmatclub.com/forum/a-triangle-has-side-lengths-of-a-b-and-c-cent...

Cheers,

Brent

## A triangle has side lengths

A triangle has side lengths of a, b, and c centimeters. Does each angle in the triangle measure less than 90 degrees?

1. The 3 semicircles whose diameters are the sides of the triangle have areas that are equal to 3 cm2, 4 cm2, and 6 cm2, respectively.

2. c < a + b < c + 2

sir i did this question in different way

rephrasing the question is the triangle acute angled triangle

side1 side2 side3

so acute angle property (side1)^2 <(side2)^2 +(side3)^2

from statement 1 we can get sides and if they satisfy the above equation or not

hence a is sufficient.

is this approach correct?

## Question link: https:/

Question link: https://gmatclub.com/forum/a-triangle-has-side-lengths-of-a-b-and-c-cent...

Yes, that approach is perfectly valid.

By the way, here's my (different) solution: https://gmatclub.com/forum/a-triangle-has-side-lengths-of-a-b-and-c-cent...

Cheers,

Brent

## In part 5.25 of your video,

## Sorry, the main purpose of

Sorry, the main purpose of that example was to show how to find the area of a triangle.

I just picked a random height to show how the formula would work.

For that particular triangle, finding the actual height would require some math that is not required for the GMAT.

Cheers,

Brent

## https://gmatclub.com/forum

If the statement (1) had said the product of BD and DC was 20, would this have been enough to answer the area of ABC?

As there's a perpendicular line BD, I assume that it has created two congruent triangles and as such DC would equal AD. Thus, in this case the area of the entire triangle would be 20

## Question link: https:/

Question link: https://gmatclub.com/forum/what-is-the-area-of-triangular-region-abc-abo...

If statement (1) had said the product of BD and DC = 20, we wouldn't have enough information to answer the area of ABC.

The presence of the perpendicular does not guarantee that AD = DC.

I drew some diagrams to reinforce this point: https://imgur.com/a/qYeN4pj

NOTE: the property you've described applies only to ISOSCELES triangles. Since we don't know whether triangle ABC is isosceles, we can't apply that property.

Cheers,

Brent

## at minute 7 can't you just

## That strategy works perfectly

That strategy works perfectly.

However, you can save some time by applying the area formula for equilateral triangles.

## Hi!

It seems that for question https://gmatclub.com/forum/in-triangle-abc-point-x-is-the-midpoint-of-side-ac-and-82671.html , there is a new concept about midpoints and similar triangles.

Is there a general rule we can remember for such questions? Thanks.

## Question link: https:/

Question link: https://gmatclub.com/forum/in-triangle-abc-point-x-is-the-midpoint-of-si...

Here are two takeaways:

Take triangle ABC

Add the midpoint of side AB and call it point X

Add the midpoint of side AC and call it point Y

Connect points X and Y

1) line segment XY will be parallel to side BC.

2) triangle AXY is similar to triangle ABC

## Thanks!

## Hi Brent,

I wanted to check with you on the approach I used for this question: https://gmatclub.com/forum/in-triangle-abc-above-if-ad-bc-what-is-the-value-of-x-347007.html.

S1: D is midpoint of side AC.

Let AD = P, so AC = 2P, and since BC = AD, then BC is also P. Since AC = 2P, BC = P, it will be a 30, 60, 90 right triangle. So angle ACB is 60 degrees. Within triangle DBC, angle ACB is 60 degrees, angle DBC = 90-x and given angle BDC is 2x, all add to 180. Solving for x, x=30.

S2. Same as S1 since it's been realized as a 30, 60, 90 right triangle.

Thank you!

## That's a good idea, but

That's a good idea, but having one side of a triangle be twice as long as one of the other sides doesn't necessarily mean the triangle is a 30-60-90 triangle.

Consider the following graphic: https://imgur.com/jrAJy7t

In the graphic, the red line is twice as long as the blue line. However, there are infinitely many triangles that can be created using those two sides. The first triangle is a 30-60-90 right triangle, but the other two are not.

So, even though it turns out that the triangle is, indeed, a 30-60-90 triangle, we need more information (in addition to one side being twice as long as another side) in order to conclude this.

I hope that helps.

## Thank you Brent, that helps!